field adjunction

Let $K$ be a field and $E$ an extension field of $K$.  If $\alpha \in E$, then the smallest subfield of $E$, that contains $K$ and $\alpha$, is denoted by $K(\alpha )$.  We say that $K(\alpha )$ is obtained from the field $K$ by adjoining the element $\alpha$ to $K$ via field adjunction.

Theorem.  $K(\alpha )$ is identical with the quotient field $Q$ of $K[\alpha ]$.

Proof. (1) Because $K[\alpha ]$ is an integral domain (as a subring of the field $E$), all possible quotients of the elements of $K[\alpha ]$ belong to $E$. So we have

$$K\cup \{\alpha \}\subseteq K[\alpha ]\subseteq Q\subseteq E,$$

and because $K(\alpha )$ was the smallest, then  $K(\alpha )\subseteq Q.$

(2) $K(\alpha )$ is a subring of $E$ containing $K$ and $\alpha$, therefore also the whole ring $K[\alpha ]$, that is,  $K[\alpha ]\subseteq K(\alpha )$.  And because $K(\alpha )$ is a field, it must contain all possible quotients of the elements of $K[\alpha ]$, i.e.,  $Q\subseteq K(\alpha )$.

In to the adjunction of one single element, we can adjoin to $K$ an arbitrary subset $S$ of $E$:  the resulting field $K(S)$ is the smallest of such subfields of $E$, i.e. the intersection of such subfields of $E$, that contain both $K$ and $S$.  We say that $K(S)$ is obtained from $K$ by adjoining the set $S$ to it.  Naturally,

$$K\subseteq K(S)\subseteq E.$$

The field $K(S)$ contains all elements of $K$ and $S$, and being a field, also all such elements that can be formed via addition, subtraction, multiplication and division from the elements of $K$ and $S$.  But such elements constitute a field, which therefore must be equal with $K(S)$.  Accordingly, we have the

Theorem.  $K(S)$ constitutes of all rational expressions formed of the elements of the field $K$ with the elements of the set $S$.

Notes.

1. $K(S)$ is the union of all fields $K(T)$ where $T$ is a finite subset of $S$.
2. $K(S_1\cup S_2)=K(S_1)(S_2)$.
3. If, especially, $S$ also is a subfield of $E$, then one may denote  $K(S)=KS$.

Title	field adjunction
Canonical name	FieldAdjunction
Date of creation	2015-02-21 15:39:45
Last modified on	2015-02-21 15:39:45
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Definition
Classification	msc 12F99
Synonym	simple extension
Related topic	GroundFieldsAndRings
Related topic	Forcing
Related topic	PolynomialRingOverFieldIsEuclideanDomain
Related topic	AConditionOfAlgebraicExtension